One begins with the empty set {}, then considers the set which contains the empty set {{}}, then the set which contains the empty set and the set which contains the empty set {{},{{}}}, and so on and so forth.
It turns out that the "and so on and so forth" is real magic and creates the ordinals and in particular large ordinals (*); This is truly mind-boggling stuff and as usual John explains it quite well.

But once you make it to the third blog post, keep in mind that "all the ordinals in this series of posts will be countable", which I find quite amazing to be honest.

(*) Initially one recreates the natural numbers as 0 = {}, 1 = {{}}, etc.
The first large ordinal is encountered as w = {0,1,2...} , i.e. the ordinal associated with the set of all natural numbers.
Then we get to w+1 = {1,2,3, ..., w} and the real fun begins...